Applied Mathematics & Optimization

, Volume 79, Issue 3, pp 695–713 | Cite as

Homogenization of Variational Inequalities for the p-Laplace Operator in Perforated Media Along Manifolds

  • D. Gómez
  • E. PérezEmail author
  • A. V. Podolskii
  • T. A. Shaposhnikova


We address homogenization problems of variational inequalities for the p-Laplace operator in a domain of \(\mathbb {R}^n\) (\(n\ge \) 3, \(p\in [2,n)\)) periodically perforated by balls of radius \(O(\varepsilon ^\alpha )\) where \(\alpha >1\) and \(\varepsilon \) is the size of the period. The perforations are distributed along a \((n-1)\)-dimensional manifold \(\gamma \), and we impose constraints for solutions and their fluxes (associated with the p-Laplacian) on the boundary of the perforations. These constraints imply that the solution is positive and that the flux is bounded from above by a negative, nonlinear monotonic function of the solution multiplied by a parameter \(\varepsilon ^{-\kappa }\), \(\kappa \in \mathbb {R}\) and \(\varepsilon \) is a small parameter that we shall make to go to zero. We analyze different relations between the parameters \(p, \, n, \, \varepsilon , \, \alpha \) and \(\kappa \), and obtain homogenized problems which are completely new in the literature even for the case \(p=2\).


Boundary homogenization Nonlinear homogenization Perforated media Variational inequality Critical relations for parameters 

Mathematics Subject Classification

35B27 35J60 35J87 35B25 


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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  • D. Gómez
    • 1
  • E. Pérez
    • 2
    Email author
  • A. V. Podolskii
    • 3
  • T. A. Shaposhnikova
    • 3
  1. 1.Departamento de Matemáticas, Estadística y ComputaciónUniversidad de CantabriaSantanderSpain
  2. 2.Departamento de Matemática Aplicada y Ciencias de la ComputaciónUniversidad de CantabriaSantanderSpain
  3. 3.Department of Differential EquationsMoscow State UniversityMoscowRussia

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